Question: Determine how many solutions exist for the system of equations. ${-6x+2y = 20}$ ${y = -6+3x}$
Explanation: Convert both equations to slope-intercept form: ${-6x+2y = 20}$ $-6x{+6x} + 2y = 20{+6x}$ $2y = 20+6x$ $y = 10+3x$ ${y = 3x+10}$ ${y = -6+3x}$ ${y = 3x-6}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = 3x+10}$ ${y = 3x-6}$ Both equations have the same slope with different y-intercepts. This means the equations are parallel. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Parallel lines never intersect, thus there are NO SOLUTIONS.